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Mathematics

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Nandakumar Ganapathy

This document discusses squares, square roots, and their properties. It includes: definitions of squares and perfect squares; tables showing the squares of integers 1-10; properties of squares and perfect squares such as tests for perfect squares; methods for finding square roots including repeated subtraction, prime factorization, and long division; and patterns in square numbers and their digital representations. Video clips are referenced to further explain some of these topics.

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SQUARES & SQUARE ROOTS BY: ROHIT KUMAR
CONTENT  SQUARES  PERFECT SQUARES  TABLE OF SQUARES  PROPERTIES OF SQUARES  PROPERTIES OF PERFECT SQUARES  PYTHAGOREAN TRIPLET  SQUARES OF INTEGERS  SQUARE ROOTS  REPEATED SUBTRACTION  PRIME FACTORISATION  LONG-DIVISION METHOD  SQUARE ROOTS OF NUMBERS IN DECIMAL FORM  PATTERN OF SQUARE NUMBER  QUICK NOTES
SQUARES In mathematics square of a number is obtained by multiplying the number by itself.  The usual notation for the formula for the square of a number n is not the product n × n, but the equivalent exponentiation n2  FOR EXAMPLE: 6 2 =6*6=36 On the next slide there is a video clipping by Adhithan who explains about SQUARES.
Perfect squares  A Perfect square is a natural number which is the square of another natural number .  For Example consider two number 84 and 36. The factors of 84 are 2*2*3*7  Factors of 36 are 2*2*3*3. The Factor of 84 cannot be grouped into pairs of identical factors. So, 84 is not a perfect. But the factor of 36 can be grouped into pairs of identical factors , like  36 = 2*2 *3*3 =62
Table of squares  NUMBERS(1 TO 10) MULTIPLICATION SQUARE NUMBER  1 1*1=12 1  2 2*2=22 4  3 3*3=32 9  4 4*4=42 16  5 5*5=52 25  6 6*6=62 36  7 7*7=72 49  8 8*8=82 64  9 9*9=92 81  10 10*10=102 100
PROPERTIES OF SQUARES The number m is a square number if and only if one can compose a square of m equal (lesser) squares: m = 12 = 1 = m = 22 = 4 = m = 32 = 9 = m = 42 = 16 = m = 52 = 25 =
PROPERTIES OF PERFECT SQUARES  A number ending in 2,3,7or 8 is never a perfect square. A number ending in an odd number of zeros is never a perfect square.  The square of even number is even.  The square of odd number is odd.  The square of a proper fraction is smaller than the fraction.  The square of a natural number ‘n’ is equal to the sum of the first ‘n’ odd numbers .  For example : n is equal to the sum of the first ‘n’ odd numbers.
Pythagorean triplet  Consider the following:-  32+42=9+16=25=52  The collection of numbers 3,4 and 5 are known as  Pythagorean triplet  For any natural number m>1, we have (2m)2+(m2-1)2 = (m2+1)2
SQUARES OF INTEGERS  Squares of negative integers:-  The square of a negative integer is always a positive integer. For example :- -m*-m=m2  -5*-5= 52 = 25  Squares of positive integers:-  The square of a positive integer is always a positive integer. For example :- m*m= m2  5* 5= 52 = 25  On the next slide there is a video clipping by Maharajan who explains about SQUARES OF INTEGERS
Square Roots In mathematics, a square root of a number x is a number y such that y2 = x ( symbol - ). For example :  There are 3 methods to find square roots , they are :-  REPEATED SUBTRACTION ( for small squares)  PRIME FACTORIZATION  LONG DIVISION  On the next slide there is a video clipping by Adhithan who explains about Square Roots
Repeated subtraction  Repeated subtraction method e.g.,- √81  Sol.:- 81-1=80  (2) 80-3=77  (3) 77-5=72  (4) 72-7=65  (5) 65-9=56  (6) 56-11=45  (7) 45-13=32  (8)32-15=17  (9) 17-17=0  Result=9 On the next slide there is a video clipping by Tarun Prasad who explains about Repeated subtraction
PRIME FACTORISATION  PRIME FACTORIZATION METHOD In order to find the square root of a perfect square , resolve it into prime factors; make pairs of similar factors , and take the product of prime factors , choosing one out of every pair.  On the next slide there is a video clipping by Tarun Prasad who explains about PRIME FACTORISATION
LONG-DIVISION METHOD  When numbers are very large , the method of finding their square roots by factorization becomes lengthy and difficult .So, we use long-division method.  For example : On the next slide there is a video clipping by Rohit Kumar who explains about LONG-DIVISION METHOD
SQUARE ROOTS OF NUMBERS IN DECIMAL FORM  For finding the square root of a decimal fraction , make the number of decimal places even by affixing a zero , if necessary; mark the periods , and find out the square root, putting the decimal point in the square root as soon as the integral part is exhausted.  For example :  On the next slide there is a video clipping by Rohit Kumar who explains about SQUARE ROOTS OF NUMBERS IN DECIMAL FORM
Pattern of square number  Pattern of square number  12 =1  112 =121  1112 =12321  11112=1234321  111112 =123454321  1111112 =12345654321  11111112 =1234567654321  111111112 =123456787654321  1111111112 =12345678987654321
QUICK NOTES  If p=m 2 , where m is a natural number, then p is a perfect square. When the sum of odd numbers is even it is a perfect square of even number and when the sum of odd numbers is odd it is a perfect square of odd numbers. To find a square root of a decimal number correct up to “n” places , we find the square root up to (n+1) places and round it off to “n” places.
Mathematics

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  • 1. SQUARES & SQUARE ROOTS BY: ROHIT KUMAR
  • 2. CONTENT  SQUARES  PERFECT SQUARES  TABLE OF SQUARES  PROPERTIES OF SQUARES  PROPERTIES OF PERFECT SQUARES  PYTHAGOREAN TRIPLET  SQUARES OF INTEGERS  SQUARE ROOTS  REPEATED SUBTRACTION  PRIME FACTORISATION  LONG-DIVISION METHOD  SQUARE ROOTS OF NUMBERS IN DECIMAL FORM  PATTERN OF SQUARE NUMBER  QUICK NOTES
  • 3. SQUARES In mathematics square of a number is obtained by multiplying the number by itself.  The usual notation for the formula for the square of a number n is not the product n × n, but the equivalent exponentiation n2  FOR EXAMPLE: 6 2 =6*6=36 On the next slide there is a video clipping by Adhithan who explains about SQUARES.
  • 4. Perfect squares  A Perfect square is a natural number which is the square of another natural number .  For Example consider two number 84 and 36. The factors of 84 are 2*2*3*7  Factors of 36 are 2*2*3*3. The Factor of 84 cannot be grouped into pairs of identical factors. So, 84 is not a perfect. But the factor of 36 can be grouped into pairs of identical factors , like  36 = 2*2 *3*3 =62
  • 5. Table of squares  NUMBERS(1 TO 10) MULTIPLICATION SQUARE NUMBER  1 1*1=12 1  2 2*2=22 4  3 3*3=32 9  4 4*4=42 16  5 5*5=52 25  6 6*6=62 36  7 7*7=72 49  8 8*8=82 64  9 9*9=92 81  10 10*10=102 100
  • 6. PROPERTIES OF SQUARES The number m is a square number if and only if one can compose a square of m equal (lesser) squares: m = 12 = 1 = m = 22 = 4 = m = 32 = 9 = m = 42 = 16 = m = 52 = 25 =
  • 7. PROPERTIES OF PERFECT SQUARES  A number ending in 2,3,7or 8 is never a perfect square. A number ending in an odd number of zeros is never a perfect square.  The square of even number is even.  The square of odd number is odd.  The square of a proper fraction is smaller than the fraction.  The square of a natural number ‘n’ is equal to the sum of the first ‘n’ odd numbers .  For example : n is equal to the sum of the first ‘n’ odd numbers.
  • 8. Pythagorean triplet  Consider the following:-  32+42=9+16=25=52  The collection of numbers 3,4 and 5 are known as  Pythagorean triplet  For any natural number m>1, we have (2m)2+(m2-1)2 = (m2+1)2
  • 9. SQUARES OF INTEGERS  Squares of negative integers:-  The square of a negative integer is always a positive integer. For example :- -m*-m=m2  -5*-5= 52 = 25  Squares of positive integers:-  The square of a positive integer is always a positive integer. For example :- m*m= m2  5* 5= 52 = 25  On the next slide there is a video clipping by Maharajan who explains about SQUARES OF INTEGERS
  • 10. Square Roots In mathematics, a square root of a number x is a number y such that y2 = x ( symbol - ). For example :  There are 3 methods to find square roots , they are :-  REPEATED SUBTRACTION ( for small squares)  PRIME FACTORIZATION  LONG DIVISION  On the next slide there is a video clipping by Adhithan who explains about Square Roots
  • 11. Repeated subtraction  Repeated subtraction method e.g.,- √81  Sol.:- 81-1=80  (2) 80-3=77  (3) 77-5=72  (4) 72-7=65  (5) 65-9=56  (6) 56-11=45  (7) 45-13=32  (8)32-15=17  (9) 17-17=0  Result=9 On the next slide there is a video clipping by Tarun Prasad who explains about Repeated subtraction
  • 12. PRIME FACTORISATION  PRIME FACTORIZATION METHOD In order to find the square root of a perfect square , resolve it into prime factors; make pairs of similar factors , and take the product of prime factors , choosing one out of every pair.  On the next slide there is a video clipping by Tarun Prasad who explains about PRIME FACTORISATION
  • 13. LONG-DIVISION METHOD  When numbers are very large , the method of finding their square roots by factorization becomes lengthy and difficult .So, we use long-division method.  For example : On the next slide there is a video clipping by Rohit Kumar who explains about LONG-DIVISION METHOD
  • 14. SQUARE ROOTS OF NUMBERS IN DECIMAL FORM  For finding the square root of a decimal fraction , make the number of decimal places even by affixing a zero , if necessary; mark the periods , and find out the square root, putting the decimal point in the square root as soon as the integral part is exhausted.  For example :  On the next slide there is a video clipping by Rohit Kumar who explains about SQUARE ROOTS OF NUMBERS IN DECIMAL FORM
  • 15. Pattern of square number  Pattern of square number  12 =1  112 =121  1112 =12321  11112=1234321  111112 =123454321  1111112 =12345654321  11111112 =1234567654321  111111112 =123456787654321  1111111112 =12345678987654321
  • 16. QUICK NOTES  If p=m 2 , where m is a natural number, then p is a perfect square. When the sum of odd numbers is even it is a perfect square of even number and when the sum of odd numbers is odd it is a perfect square of odd numbers. To find a square root of a decimal number correct up to “n” places , we find the square root up to (n+1) places and round it off to “n” places.